Existence of a solution to a coupled elliptic system
type: Item
instance of: scholarly article
title: Existence of a solution to a coupled elliptic system
zbMATH Open document ID: 0791.35043
DOI: 10.1016/0893-9659(94)90030-2
author: Raphaèle Herbin
author: T. Gallouët
published in: Applied Mathematics Letters
publication date: 1994-07-18T00:00:00Z
review text: Suppose \(\Omega\) is a bounded regular domain of \(\mathbb{R}^ N\), \(N>1\), \(\partial\Omega\) its boundary, \(\sigma\), \(\lambda\) and \(f\) are bounded functions from \(\Omega\times\mathbb{R}\) to \(\mathbb{R}\), continuous with respect to \(y\in\mathbb{R}\) for a.e. \(x\in\Omega\), and measurable with respect to \(x\in\Omega\) for any \(y\in\mathbb{R}\), and such that: \[ \exists\alpha>0;\quad \alpha\leq \sigma(\cdot,y) \text{ and } \alpha\leq \lambda(\cdot,y), \qquad \forall y\in\mathbb{R}, \quad \text{for a.e. } x\in\Omega. \] The aim of this work is to prove the following existence theorem: Theorem. Under the above assumptions, there exists a solution \((u,\varphi)\) satisfying: \[ u\in \textstyle {\bigcap_{p<{N\over {N-1}}}} W_ 0^{1,p}(\Omega), \qquad \varphi\in H_ 0^ 1(\Omega) \] \[ \begin{aligned} \int_ \Omega &\sigma (x,u(x)) D\varphi(x) D\psi(x)dx= \int_ \Omega f(x,u(x))\psi(x)dx,\qquad \forall\psi\in H_ 0^ 1 (\Omega),\\ \int_ \Omega &\lambda(x,u(x)) Du(x) Dv(x)dx= \int_ \Omega \sigma(x,u(x)) D\varphi(x) D\varphi(x)v(x)dx, \quad \forall v\in \bigcup_{q>N} W_ 0^{1,q}(\Omega).\end{aligned} \] {}.
Mathematics Subject Classification ID: 35J65
Mathematics Subject Classification ID: 35D05
zbMATH DE Number: 571547
zbMATH Keywords: applications to electro-chemistry
zbMATH Keywords: quasilinear elliptic system
zbMATH Keywords: Schauder's fixed point theorem
zbMATH Keywords: bounded regular domain
MaRDI profile type: Publication
full work available at URL: 0893-9659(94)90030-2
OpenAlex ID: W2046535389
cites work: NUMERICAL MODELING OF INDUCTION HEATING FOR TWO-DIMENSIONAL GEOMETRIES
cites work: Nonlinear elliptic and parabolic equations involving measure data
cites work: Q5518040
cites work: Q4197852
description: scientific article; zbMATH DE number 571547